A Unified Algebraic Approach to 2-D and 3-D Motion Segmentation

A Unified Algebraic Approach to 2-D and 3-D Motion Segmentation

2006. 6. 9 (Fri)Young Ki Baik, Computer Vision Lab.

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A Unified Algebraic Approach to 2D and 3D Motion Segmentation

• References• A Unified Algebraic Approach to 2-D and 3-

D Motion Segmentation• Rene Vidal and Yi Ma (ECCV 2004)

• Generalized Principal Component Analysis (GPCA)

• Rene Vidal, Yi Ma, et. al. (PAMI 2005)

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• Contents• Introduction• GPCA• 2-D motion segmentation• 3-D motion segmentation• Experimental results• Summary

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• Introduction (Motion segmentation)• Target

• 2D, 3D motion segmentation

• Problem statement• Most previous work

Iterative approach (EM, RANSAC, etc.) Manual or random initial value.

Cause of divergence or bad results

• Proposed algorithm• Good initial value using non-iterative

algebraic method

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• Introduction (This paper)• Contribution

• Applying GPCA to 2D, 3D motion segmentation problem

• Condition• Subspaces are all linear.• Known correspondences• Known number of subspace (class)

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• GPCA (Generalized PCA)• GPCA treats heterogeneous data and

multiple subset with different linear model.

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• GPCA (Generalized PCA)• Find the basis of each subspace which

orthogonal to data x

b

x

SforT xbx 0

SdatabasissubspaceS

:::

xb

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• GPCA (Generalized PCA)• A homogeneous polynomial of degree n

• The mixture of subspaces can be linearly fitting general polynomial to the given data.

• Example• Number of subspace n = 2• Vector size k = 3

xcx nT

n vp mapVeronesev

polynomialpsetdata

subspaceofnumbern

n

n

::

: :

x

321 ,, xxxx

0236325

224313212

2112 xcxxcxcxxcxxcxcxp

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• GPCA (Generalized PCA)• Find the basis from derivatives of the

polynomials with y on the S

b

y

by ~nDp

S databasissubspaceS

:::

yb

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• GPCA (Algorithm)• Fitting polynomials to data lying in multiple

subspaces

• Obtaining basis of each subspace by polynomial differentiation

• Choosing data per subspace by using basis

by ~nDp

xcx nT

n vp 0 Ac

SforT xbx 0

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• Motion segmentation• Notation

• Let be a vector in or .z KR KC

n

in

TTin vp

1

zczbz

zKnN

vector ofdimension :subspace ofnumber :data ofnumber :

mapVeronesev

polymonialppofvectortcoefficien

subspacethiofbasis

n

n

n

i

::

: :

cb

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• 2D Motion segmentation• Translation case

• Under 2-d translation motion model, the two images are related by on out of n possible 2-d translation .

iTxx 12

222

21 ,, RTRxRx i

nii 12

RT

2

12

01

1 Cxx

Tzb

iTi

2212 , CTCxx i

z

1

231

21

13



• Finding coefficient c (fitting polynomial)

jbaj011

12 xxz

0201 622

5432122 abjcbacjbcacjccvp zcz

06 cAN

z2psvd

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• Finding basis (using polynomial differentiation)

zzz 2

2pDp

i

Dpi yzzb

2~ 0iTi yb

zz

z n

n

alln Dp

py

minarg 0 lastI

1

1

ImRe

i

ii b

bT

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• Segmentation

njjTn

allSn zzb

n minargN~1 i for

end

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• 2D Motion segmentation• Affine motion case

• In this case, we assume that the images are related by a collection on n 2-D affine motion models .

111

232221

13121112

xxAx

aaaaaa

i

nii 132

RA

111

2313221221111

2

xxx jaajaajaaaTi

01

1

2

12

11

x

xx

zb Ti

Ti a

4Cz

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• 3D Motion segmentation• Epipolar constraint • Homography

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• Experimental results

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• Summary• Contribution

• Applying GPCA to 2D, 3D motion segmentation problem

• Good initial value using non-iterative algebraic method

• Limitation• Linear subspace• Known correspondences and number of subspace • If subspace is increased, then computational

complexity will be exponentially increased.

A Unified Algebraic Approach to 2-D and 3-D Motion Segmentation

Documents

d motion segmentation2006

d translation motion

d affine motion models

iterative algebraic

heterogeneous data

given data

examplenumber of subspace

fitting general polynomial